Context. Gaia’s very accurate astrometric measurements
will allow the optical realisation of the International Celestial Reference System to be
improved by a few orders of magnitude. Several sets of quasars are used to define a
kinematically stable non-rotating reference frame with the barycentre of the solar system
as its origin. Gaia will also observe a large number of galaxies.
Although they are not point-like, it may be possible to determine accurate positions and
proper motions for some of their compact bright features.

Aims. The optical stability of the quasars is critical, and we
investigate how accurately the reference frame can be recovered. Various proper motion
patterns are also present in the data, the best known is caused by the acceleration of the
solar system barycentre, presumably, towards the Galactic centre. We review some other
less well-known effects that are not part of standard astrometric models.

Methods. We modelled quasars and galaxies using realistic sky
distributions, magnitudes, and redshifts. Position variability was introduced using a
Markov chain model. The reference frame was determined using the algorithm developed for
the Gaia mission, which also determines the acceleration of the solar
system. We also tested a method for measuring the velocity of the solar system barycentre
in a cosmological frame.

Results. We simulated the recovery of the reference frame and the
acceleration of the solar system and conclude that they are not significantly disturbed by
quasar variability, which is statistically averaged. However, the effect of a non-uniform
sky distribution of the quasars can result in a correlation between the parameters
describing the spin components of the reference frame and the acceleration components,
which degrades the solution. Our results suggest that an attempt should be made to
astrometrically determine the redshift-dependent apparent drift of galaxies that is due to
our velocity relative to the cosmic microwave background, which in principle could allow
determining the Hubble parameter.

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